Solve 2/1-sqrt{3}

$$\frac { 2 } { 1 - \sqrt { 3 } }$$

Evaluate

$$-\sqrt{3}-1\approx -2.732050808$$

Solution Steps

Rationalize the denominator of \frac{2}{1-\sqrt{3}} by multiplying numerator and denominator by 1+\sqrt{3}.

$$\frac{2\left(1+\sqrt{3}\right)}{\left(1-\sqrt{3}\right)\left(1+\sqrt{3}\right)}$$

Consider \left(1-\sqrt{3}\right)\left(1+\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{2\left(1+\sqrt{3}\right)}{1^{2}-\left(\sqrt{3}\right)^{2}}$$

Square 1. Square \sqrt{3}.

$$\frac{2\left(1+\sqrt{3}\right)}{1-3}$$

Subtract 3 from 1 to get -2.

$$\frac{2\left(1+\sqrt{3}\right)}{-2}$$

Cancel out -2 and -2.

$$-\left(1+\sqrt{3}\right)$$

To find the opposite of 1+\sqrt{3}, find the opposite of each term.

$$-1-\sqrt{3}$$